4.10.42 \((5 y(x)+3 x+6) y'(x)=7 y(x)+x+2\)

ODE
\[ (5 y(x)+3 x+6) y'(x)=7 y(x)+x+2 \] ODE Classification

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type``class A`]]

Book solution method
Equation linear in the variables, \(y'(x)=f\left (\frac {X_1}{X_2} \right ) \)

Mathematica
cpu = 0.355231 (sec), leaf count = 4961

\[\left \{\left \{y(x)\to \frac {1}{5} \left (\frac {1}{\text {Root}\left [\left (16777216 e^{\frac {24 c_1}{25}} x^8+268435456 e^{\frac {24 c_1}{25}} x^7+1879048192 e^{\frac {24 c_1}{25}} x^6+7516192768 e^{\frac {24 c_1}{25}} x^5+18790481920 e^{\frac {24 c_1}{25}} x^4+30064771072 e^{\frac {24 c_1}{25}} x^3+30064771072 e^{\frac {24 c_1}{25}} x^2+16777216 x^2+17179869184 e^{\frac {24 c_1}{25}} x+67108864 x+4294967296 e^{\frac {24 c_1}{25}}+67108864\right ) \text {$\#$1}^8+\left (-16777216 e^{\frac {24 c_1}{25}} x^7-234881024 e^{\frac {24 c_1}{25}} x^6-1409286144 e^{\frac {24 c_1}{25}} x^5-4697620480 e^{\frac {24 c_1}{25}} x^4-9395240960 e^{\frac {24 c_1}{25}} x^3-11274289152 e^{\frac {24 c_1}{25}} x^2-7516192768 e^{\frac {24 c_1}{25}} x-16777216 x-2147483648 e^{\frac {24 c_1}{25}}-33554432\right ) \text {$\#$1}^7+\left (7340032 e^{\frac {24 c_1}{25}} x^6+88080384 e^{\frac {24 c_1}{25}} x^5+440401920 e^{\frac {24 c_1}{25}} x^4+1174405120 e^{\frac {24 c_1}{25}} x^3+1761607680 e^{\frac {24 c_1}{25}} x^2+1409286144 e^{\frac {24 c_1}{25}} x+469762048 e^{\frac {24 c_1}{25}}+4194304\right ) \text {$\#$1}^6+\left (-1835008 e^{\frac {24 c_1}{25}} x^5-18350080 e^{\frac {24 c_1}{25}} x^4-73400320 e^{\frac {24 c_1}{25}} x^3-146800640 e^{\frac {24 c_1}{25}} x^2-146800640 e^{\frac {24 c_1}{25}} x-58720256 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^5+\left (286720 e^{\frac {24 c_1}{25}} x^4+2293760 e^{\frac {24 c_1}{25}} x^3+6881280 e^{\frac {24 c_1}{25}} x^2+9175040 e^{\frac {24 c_1}{25}} x+4587520 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^4+\left (-28672 e^{\frac {24 c_1}{25}} x^3-172032 e^{\frac {24 c_1}{25}} x^2-344064 e^{\frac {24 c_1}{25}} x-229376 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^3+\left (1792 e^{\frac {24 c_1}{25}} x^2+7168 e^{\frac {24 c_1}{25}} x+7168 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^2+\left (-64 e^{\frac {24 c_1}{25}} x-128 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}+e^{\frac {24 c_1}{25}}\& ,1\right ]}-3 (x+2)\right )\right \},\left \{y(x)\to \frac {1}{5} \left (\frac {1}{\text {Root}\left [\left (16777216 e^{\frac {24 c_1}{25}} x^8+268435456 e^{\frac {24 c_1}{25}} x^7+1879048192 e^{\frac {24 c_1}{25}} x^6+7516192768 e^{\frac {24 c_1}{25}} x^5+18790481920 e^{\frac {24 c_1}{25}} x^4+30064771072 e^{\frac {24 c_1}{25}} x^3+30064771072 e^{\frac {24 c_1}{25}} x^2+16777216 x^2+17179869184 e^{\frac {24 c_1}{25}} x+67108864 x+4294967296 e^{\frac {24 c_1}{25}}+67108864\right ) \text {$\#$1}^8+\left (-16777216 e^{\frac {24 c_1}{25}} x^7-234881024 e^{\frac {24 c_1}{25}} x^6-1409286144 e^{\frac {24 c_1}{25}} x^5-4697620480 e^{\frac {24 c_1}{25}} x^4-9395240960 e^{\frac {24 c_1}{25}} x^3-11274289152 e^{\frac {24 c_1}{25}} x^2-7516192768 e^{\frac {24 c_1}{25}} x-16777216 x-2147483648 e^{\frac {24 c_1}{25}}-33554432\right ) \text {$\#$1}^7+\left (7340032 e^{\frac {24 c_1}{25}} x^6+88080384 e^{\frac {24 c_1}{25}} x^5+440401920 e^{\frac {24 c_1}{25}} x^4+1174405120 e^{\frac {24 c_1}{25}} x^3+1761607680 e^{\frac {24 c_1}{25}} x^2+1409286144 e^{\frac {24 c_1}{25}} x+469762048 e^{\frac {24 c_1}{25}}+4194304\right ) \text {$\#$1}^6+\left (-1835008 e^{\frac {24 c_1}{25}} x^5-18350080 e^{\frac {24 c_1}{25}} x^4-73400320 e^{\frac {24 c_1}{25}} x^3-146800640 e^{\frac {24 c_1}{25}} x^2-146800640 e^{\frac {24 c_1}{25}} x-58720256 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^5+\left (286720 e^{\frac {24 c_1}{25}} x^4+2293760 e^{\frac {24 c_1}{25}} x^3+6881280 e^{\frac {24 c_1}{25}} x^2+9175040 e^{\frac {24 c_1}{25}} x+4587520 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^4+\left (-28672 e^{\frac {24 c_1}{25}} x^3-172032 e^{\frac {24 c_1}{25}} x^2-344064 e^{\frac {24 c_1}{25}} x-229376 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^3+\left (1792 e^{\frac {24 c_1}{25}} x^2+7168 e^{\frac {24 c_1}{25}} x+7168 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^2+\left (-64 e^{\frac {24 c_1}{25}} x-128 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}+e^{\frac {24 c_1}{25}}\& ,2\right ]}-3 (x+2)\right )\right \},\left \{y(x)\to \frac {1}{5} \left (\frac {1}{\text {Root}\left [\left (16777216 e^{\frac {24 c_1}{25}} x^8+268435456 e^{\frac {24 c_1}{25}} x^7+1879048192 e^{\frac {24 c_1}{25}} x^6+7516192768 e^{\frac {24 c_1}{25}} x^5+18790481920 e^{\frac {24 c_1}{25}} x^4+30064771072 e^{\frac {24 c_1}{25}} x^3+30064771072 e^{\frac {24 c_1}{25}} x^2+16777216 x^2+17179869184 e^{\frac {24 c_1}{25}} x+67108864 x+4294967296 e^{\frac {24 c_1}{25}}+67108864\right ) \text {$\#$1}^8+\left (-16777216 e^{\frac {24 c_1}{25}} x^7-234881024 e^{\frac {24 c_1}{25}} x^6-1409286144 e^{\frac {24 c_1}{25}} x^5-4697620480 e^{\frac {24 c_1}{25}} x^4-9395240960 e^{\frac {24 c_1}{25}} x^3-11274289152 e^{\frac {24 c_1}{25}} x^2-7516192768 e^{\frac {24 c_1}{25}} x-16777216 x-2147483648 e^{\frac {24 c_1}{25}}-33554432\right ) \text {$\#$1}^7+\left (7340032 e^{\frac {24 c_1}{25}} x^6+88080384 e^{\frac {24 c_1}{25}} x^5+440401920 e^{\frac {24 c_1}{25}} x^4+1174405120 e^{\frac {24 c_1}{25}} x^3+1761607680 e^{\frac {24 c_1}{25}} x^2+1409286144 e^{\frac {24 c_1}{25}} x+469762048 e^{\frac {24 c_1}{25}}+4194304\right ) \text {$\#$1}^6+\left (-1835008 e^{\frac {24 c_1}{25}} x^5-18350080 e^{\frac {24 c_1}{25}} x^4-73400320 e^{\frac {24 c_1}{25}} x^3-146800640 e^{\frac {24 c_1}{25}} x^2-146800640 e^{\frac {24 c_1}{25}} x-58720256 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^5+\left (286720 e^{\frac {24 c_1}{25}} x^4+2293760 e^{\frac {24 c_1}{25}} x^3+6881280 e^{\frac {24 c_1}{25}} x^2+9175040 e^{\frac {24 c_1}{25}} x+4587520 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^4+\left (-28672 e^{\frac {24 c_1}{25}} x^3-172032 e^{\frac {24 c_1}{25}} x^2-344064 e^{\frac {24 c_1}{25}} x-229376 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^3+\left (1792 e^{\frac {24 c_1}{25}} x^2+7168 e^{\frac {24 c_1}{25}} x+7168 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^2+\left (-64 e^{\frac {24 c_1}{25}} x-128 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}+e^{\frac {24 c_1}{25}}\& ,3\right ]}-3 (x+2)\right )\right \},\left \{y(x)\to \frac {1}{5} \left (\frac {1}{\text {Root}\left [\left (16777216 e^{\frac {24 c_1}{25}} x^8+268435456 e^{\frac {24 c_1}{25}} x^7+1879048192 e^{\frac {24 c_1}{25}} x^6+7516192768 e^{\frac {24 c_1}{25}} x^5+18790481920 e^{\frac {24 c_1}{25}} x^4+30064771072 e^{\frac {24 c_1}{25}} x^3+30064771072 e^{\frac {24 c_1}{25}} x^2+16777216 x^2+17179869184 e^{\frac {24 c_1}{25}} x+67108864 x+4294967296 e^{\frac {24 c_1}{25}}+67108864\right ) \text {$\#$1}^8+\left (-16777216 e^{\frac {24 c_1}{25}} x^7-234881024 e^{\frac {24 c_1}{25}} x^6-1409286144 e^{\frac {24 c_1}{25}} x^5-4697620480 e^{\frac {24 c_1}{25}} x^4-9395240960 e^{\frac {24 c_1}{25}} x^3-11274289152 e^{\frac {24 c_1}{25}} x^2-7516192768 e^{\frac {24 c_1}{25}} x-16777216 x-2147483648 e^{\frac {24 c_1}{25}}-33554432\right ) \text {$\#$1}^7+\left (7340032 e^{\frac {24 c_1}{25}} x^6+88080384 e^{\frac {24 c_1}{25}} x^5+440401920 e^{\frac {24 c_1}{25}} x^4+1174405120 e^{\frac {24 c_1}{25}} x^3+1761607680 e^{\frac {24 c_1}{25}} x^2+1409286144 e^{\frac {24 c_1}{25}} x+469762048 e^{\frac {24 c_1}{25}}+4194304\right ) \text {$\#$1}^6+\left (-1835008 e^{\frac {24 c_1}{25}} x^5-18350080 e^{\frac {24 c_1}{25}} x^4-73400320 e^{\frac {24 c_1}{25}} x^3-146800640 e^{\frac {24 c_1}{25}} x^2-146800640 e^{\frac {24 c_1}{25}} x-58720256 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^5+\left (286720 e^{\frac {24 c_1}{25}} x^4+2293760 e^{\frac {24 c_1}{25}} x^3+6881280 e^{\frac {24 c_1}{25}} x^2+9175040 e^{\frac {24 c_1}{25}} x+4587520 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^4+\left (-28672 e^{\frac {24 c_1}{25}} x^3-172032 e^{\frac {24 c_1}{25}} x^2-344064 e^{\frac {24 c_1}{25}} x-229376 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^3+\left (1792 e^{\frac {24 c_1}{25}} x^2+7168 e^{\frac {24 c_1}{25}} x+7168 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^2+\left (-64 e^{\frac {24 c_1}{25}} x-128 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}+e^{\frac {24 c_1}{25}}\& ,4\right ]}-3 (x+2)\right )\right \},\left \{y(x)\to \frac {1}{5} \left (\frac {1}{\text {Root}\left [\left (16777216 e^{\frac {24 c_1}{25}} x^8+268435456 e^{\frac {24 c_1}{25}} x^7+1879048192 e^{\frac {24 c_1}{25}} x^6+7516192768 e^{\frac {24 c_1}{25}} x^5+18790481920 e^{\frac {24 c_1}{25}} x^4+30064771072 e^{\frac {24 c_1}{25}} x^3+30064771072 e^{\frac {24 c_1}{25}} x^2+16777216 x^2+17179869184 e^{\frac {24 c_1}{25}} x+67108864 x+4294967296 e^{\frac {24 c_1}{25}}+67108864\right ) \text {$\#$1}^8+\left (-16777216 e^{\frac {24 c_1}{25}} x^7-234881024 e^{\frac {24 c_1}{25}} x^6-1409286144 e^{\frac {24 c_1}{25}} x^5-4697620480 e^{\frac {24 c_1}{25}} x^4-9395240960 e^{\frac {24 c_1}{25}} x^3-11274289152 e^{\frac {24 c_1}{25}} x^2-7516192768 e^{\frac {24 c_1}{25}} x-16777216 x-2147483648 e^{\frac {24 c_1}{25}}-33554432\right ) \text {$\#$1}^7+\left (7340032 e^{\frac {24 c_1}{25}} x^6+88080384 e^{\frac {24 c_1}{25}} x^5+440401920 e^{\frac {24 c_1}{25}} x^4+1174405120 e^{\frac {24 c_1}{25}} x^3+1761607680 e^{\frac {24 c_1}{25}} x^2+1409286144 e^{\frac {24 c_1}{25}} x+469762048 e^{\frac {24 c_1}{25}}+4194304\right ) \text {$\#$1}^6+\left (-1835008 e^{\frac {24 c_1}{25}} x^5-18350080 e^{\frac {24 c_1}{25}} x^4-73400320 e^{\frac {24 c_1}{25}} x^3-146800640 e^{\frac {24 c_1}{25}} x^2-146800640 e^{\frac {24 c_1}{25}} x-58720256 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^5+\left (286720 e^{\frac {24 c_1}{25}} x^4+2293760 e^{\frac {24 c_1}{25}} x^3+6881280 e^{\frac {24 c_1}{25}} x^2+9175040 e^{\frac {24 c_1}{25}} x+4587520 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^4+\left (-28672 e^{\frac {24 c_1}{25}} x^3-172032 e^{\frac {24 c_1}{25}} x^2-344064 e^{\frac {24 c_1}{25}} x-229376 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^3+\left (1792 e^{\frac {24 c_1}{25}} x^2+7168 e^{\frac {24 c_1}{25}} x+7168 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^2+\left (-64 e^{\frac {24 c_1}{25}} x-128 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}+e^{\frac {24 c_1}{25}}\& ,5\right ]}-3 (x+2)\right )\right \},\left \{y(x)\to \frac {1}{5} \left (\frac {1}{\text {Root}\left [\left (16777216 e^{\frac {24 c_1}{25}} x^8+268435456 e^{\frac {24 c_1}{25}} x^7+1879048192 e^{\frac {24 c_1}{25}} x^6+7516192768 e^{\frac {24 c_1}{25}} x^5+18790481920 e^{\frac {24 c_1}{25}} x^4+30064771072 e^{\frac {24 c_1}{25}} x^3+30064771072 e^{\frac {24 c_1}{25}} x^2+16777216 x^2+17179869184 e^{\frac {24 c_1}{25}} x+67108864 x+4294967296 e^{\frac {24 c_1}{25}}+67108864\right ) \text {$\#$1}^8+\left (-16777216 e^{\frac {24 c_1}{25}} x^7-234881024 e^{\frac {24 c_1}{25}} x^6-1409286144 e^{\frac {24 c_1}{25}} x^5-4697620480 e^{\frac {24 c_1}{25}} x^4-9395240960 e^{\frac {24 c_1}{25}} x^3-11274289152 e^{\frac {24 c_1}{25}} x^2-7516192768 e^{\frac {24 c_1}{25}} x-16777216 x-2147483648 e^{\frac {24 c_1}{25}}-33554432\right ) \text {$\#$1}^7+\left (7340032 e^{\frac {24 c_1}{25}} x^6+88080384 e^{\frac {24 c_1}{25}} x^5+440401920 e^{\frac {24 c_1}{25}} x^4+1174405120 e^{\frac {24 c_1}{25}} x^3+1761607680 e^{\frac {24 c_1}{25}} x^2+1409286144 e^{\frac {24 c_1}{25}} x+469762048 e^{\frac {24 c_1}{25}}+4194304\right ) \text {$\#$1}^6+\left (-1835008 e^{\frac {24 c_1}{25}} x^5-18350080 e^{\frac {24 c_1}{25}} x^4-73400320 e^{\frac {24 c_1}{25}} x^3-146800640 e^{\frac {24 c_1}{25}} x^2-146800640 e^{\frac {24 c_1}{25}} x-58720256 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^5+\left (286720 e^{\frac {24 c_1}{25}} x^4+2293760 e^{\frac {24 c_1}{25}} x^3+6881280 e^{\frac {24 c_1}{25}} x^2+9175040 e^{\frac {24 c_1}{25}} x+4587520 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^4+\left (-28672 e^{\frac {24 c_1}{25}} x^3-172032 e^{\frac {24 c_1}{25}} x^2-344064 e^{\frac {24 c_1}{25}} x-229376 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^3+\left (1792 e^{\frac {24 c_1}{25}} x^2+7168 e^{\frac {24 c_1}{25}} x+7168 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^2+\left (-64 e^{\frac {24 c_1}{25}} x-128 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}+e^{\frac {24 c_1}{25}}\& ,6\right ]}-3 (x+2)\right )\right \},\left \{y(x)\to \frac {1}{5} \left (\frac {1}{\text {Root}\left [\left (16777216 e^{\frac {24 c_1}{25}} x^8+268435456 e^{\frac {24 c_1}{25}} x^7+1879048192 e^{\frac {24 c_1}{25}} x^6+7516192768 e^{\frac {24 c_1}{25}} x^5+18790481920 e^{\frac {24 c_1}{25}} x^4+30064771072 e^{\frac {24 c_1}{25}} x^3+30064771072 e^{\frac {24 c_1}{25}} x^2+16777216 x^2+17179869184 e^{\frac {24 c_1}{25}} x+67108864 x+4294967296 e^{\frac {24 c_1}{25}}+67108864\right ) \text {$\#$1}^8+\left (-16777216 e^{\frac {24 c_1}{25}} x^7-234881024 e^{\frac {24 c_1}{25}} x^6-1409286144 e^{\frac {24 c_1}{25}} x^5-4697620480 e^{\frac {24 c_1}{25}} x^4-9395240960 e^{\frac {24 c_1}{25}} x^3-11274289152 e^{\frac {24 c_1}{25}} x^2-7516192768 e^{\frac {24 c_1}{25}} x-16777216 x-2147483648 e^{\frac {24 c_1}{25}}-33554432\right ) \text {$\#$1}^7+\left (7340032 e^{\frac {24 c_1}{25}} x^6+88080384 e^{\frac {24 c_1}{25}} x^5+440401920 e^{\frac {24 c_1}{25}} x^4+1174405120 e^{\frac {24 c_1}{25}} x^3+1761607680 e^{\frac {24 c_1}{25}} x^2+1409286144 e^{\frac {24 c_1}{25}} x+469762048 e^{\frac {24 c_1}{25}}+4194304\right ) \text {$\#$1}^6+\left (-1835008 e^{\frac {24 c_1}{25}} x^5-18350080 e^{\frac {24 c_1}{25}} x^4-73400320 e^{\frac {24 c_1}{25}} x^3-146800640 e^{\frac {24 c_1}{25}} x^2-146800640 e^{\frac {24 c_1}{25}} x-58720256 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^5+\left (286720 e^{\frac {24 c_1}{25}} x^4+2293760 e^{\frac {24 c_1}{25}} x^3+6881280 e^{\frac {24 c_1}{25}} x^2+9175040 e^{\frac {24 c_1}{25}} x+4587520 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^4+\left (-28672 e^{\frac {24 c_1}{25}} x^3-172032 e^{\frac {24 c_1}{25}} x^2-344064 e^{\frac {24 c_1}{25}} x-229376 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^3+\left (1792 e^{\frac {24 c_1}{25}} x^2+7168 e^{\frac {24 c_1}{25}} x+7168 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^2+\left (-64 e^{\frac {24 c_1}{25}} x-128 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}+e^{\frac {24 c_1}{25}}\& ,7\right ]}-3 (x+2)\right )\right \},\left \{y(x)\to \frac {1}{5} \left (\frac {1}{\text {Root}\left [\left (16777216 e^{\frac {24 c_1}{25}} x^8+268435456 e^{\frac {24 c_1}{25}} x^7+1879048192 e^{\frac {24 c_1}{25}} x^6+7516192768 e^{\frac {24 c_1}{25}} x^5+18790481920 e^{\frac {24 c_1}{25}} x^4+30064771072 e^{\frac {24 c_1}{25}} x^3+30064771072 e^{\frac {24 c_1}{25}} x^2+16777216 x^2+17179869184 e^{\frac {24 c_1}{25}} x+67108864 x+4294967296 e^{\frac {24 c_1}{25}}+67108864\right ) \text {$\#$1}^8+\left (-16777216 e^{\frac {24 c_1}{25}} x^7-234881024 e^{\frac {24 c_1}{25}} x^6-1409286144 e^{\frac {24 c_1}{25}} x^5-4697620480 e^{\frac {24 c_1}{25}} x^4-9395240960 e^{\frac {24 c_1}{25}} x^3-11274289152 e^{\frac {24 c_1}{25}} x^2-7516192768 e^{\frac {24 c_1}{25}} x-16777216 x-2147483648 e^{\frac {24 c_1}{25}}-33554432\right ) \text {$\#$1}^7+\left (7340032 e^{\frac {24 c_1}{25}} x^6+88080384 e^{\frac {24 c_1}{25}} x^5+440401920 e^{\frac {24 c_1}{25}} x^4+1174405120 e^{\frac {24 c_1}{25}} x^3+1761607680 e^{\frac {24 c_1}{25}} x^2+1409286144 e^{\frac {24 c_1}{25}} x+469762048 e^{\frac {24 c_1}{25}}+4194304\right ) \text {$\#$1}^6+\left (-1835008 e^{\frac {24 c_1}{25}} x^5-18350080 e^{\frac {24 c_1}{25}} x^4-73400320 e^{\frac {24 c_1}{25}} x^3-146800640 e^{\frac {24 c_1}{25}} x^2-146800640 e^{\frac {24 c_1}{25}} x-58720256 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^5+\left (286720 e^{\frac {24 c_1}{25}} x^4+2293760 e^{\frac {24 c_1}{25}} x^3+6881280 e^{\frac {24 c_1}{25}} x^2+9175040 e^{\frac {24 c_1}{25}} x+4587520 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^4+\left (-28672 e^{\frac {24 c_1}{25}} x^3-172032 e^{\frac {24 c_1}{25}} x^2-344064 e^{\frac {24 c_1}{25}} x-229376 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^3+\left (1792 e^{\frac {24 c_1}{25}} x^2+7168 e^{\frac {24 c_1}{25}} x+7168 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^2+\left (-64 e^{\frac {24 c_1}{25}} x-128 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}+e^{\frac {24 c_1}{25}}\& ,8\right ]}-3 (x+2)\right )\right \}\right \}\]

Maple
cpu = 0.03 (sec), leaf count = 47

\[ \left \{ -{\frac {4}{3}\ln \left ({\frac {-y \relax (x ) +2+x}{2+x}} \right ) }+{\frac {1}{3}\ln \left ({\frac {-5\,y \relax (x ) -2-x}{2+x}} \right ) }-\ln \left (2+x \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[(6 + 3*x + 5*y[x])*y'[x] == 2 + x + 7*y[x],y[x],x]

Mathematica raw output

{{y[x] -> (-3*(2 + x) + Root[E^((24*C[1])/25) + (-128*E^((24*C[1])/25) - 64*E^((
24*C[1])/25)*x)*#1 + (7168*E^((24*C[1])/25) + 7168*E^((24*C[1])/25)*x + 1792*E^(
(24*C[1])/25)*x^2)*#1^2 + (-229376*E^((24*C[1])/25) - 344064*E^((24*C[1])/25)*x 
- 172032*E^((24*C[1])/25)*x^2 - 28672*E^((24*C[1])/25)*x^3)*#1^3 + (4587520*E^((
24*C[1])/25) + 9175040*E^((24*C[1])/25)*x + 6881280*E^((24*C[1])/25)*x^2 + 22937
60*E^((24*C[1])/25)*x^3 + 286720*E^((24*C[1])/25)*x^4)*#1^4 + (-58720256*E^((24*
C[1])/25) - 146800640*E^((24*C[1])/25)*x - 146800640*E^((24*C[1])/25)*x^2 - 7340
0320*E^((24*C[1])/25)*x^3 - 18350080*E^((24*C[1])/25)*x^4 - 1835008*E^((24*C[1])
/25)*x^5)*#1^5 + (4194304 + 469762048*E^((24*C[1])/25) + 1409286144*E^((24*C[1])
/25)*x + 1761607680*E^((24*C[1])/25)*x^2 + 1174405120*E^((24*C[1])/25)*x^3 + 440
401920*E^((24*C[1])/25)*x^4 + 88080384*E^((24*C[1])/25)*x^5 + 7340032*E^((24*C[1
])/25)*x^6)*#1^6 + (-33554432 - 2147483648*E^((24*C[1])/25) - 16777216*x - 75161
92768*E^((24*C[1])/25)*x - 11274289152*E^((24*C[1])/25)*x^2 - 9395240960*E^((24*
C[1])/25)*x^3 - 4697620480*E^((24*C[1])/25)*x^4 - 1409286144*E^((24*C[1])/25)*x^
5 - 234881024*E^((24*C[1])/25)*x^6 - 16777216*E^((24*C[1])/25)*x^7)*#1^7 + (6710
8864 + 4294967296*E^((24*C[1])/25) + 67108864*x + 17179869184*E^((24*C[1])/25)*x
 + 16777216*x^2 + 30064771072*E^((24*C[1])/25)*x^2 + 30064771072*E^((24*C[1])/25
)*x^3 + 18790481920*E^((24*C[1])/25)*x^4 + 7516192768*E^((24*C[1])/25)*x^5 + 187
9048192*E^((24*C[1])/25)*x^6 + 268435456*E^((24*C[1])/25)*x^7 + 16777216*E^((24*
C[1])/25)*x^8)*#1^8 & , 1]^(-1))/5}, {y[x] -> (-3*(2 + x) + Root[E^((24*C[1])/25
) + (-128*E^((24*C[1])/25) - 64*E^((24*C[1])/25)*x)*#1 + (7168*E^((24*C[1])/25) 
+ 7168*E^((24*C[1])/25)*x + 1792*E^((24*C[1])/25)*x^2)*#1^2 + (-229376*E^((24*C[
1])/25) - 344064*E^((24*C[1])/25)*x - 172032*E^((24*C[1])/25)*x^2 - 28672*E^((24
*C[1])/25)*x^3)*#1^3 + (4587520*E^((24*C[1])/25) + 9175040*E^((24*C[1])/25)*x + 
6881280*E^((24*C[1])/25)*x^2 + 2293760*E^((24*C[1])/25)*x^3 + 286720*E^((24*C[1]
)/25)*x^4)*#1^4 + (-58720256*E^((24*C[1])/25) - 146800640*E^((24*C[1])/25)*x - 1
46800640*E^((24*C[1])/25)*x^2 - 73400320*E^((24*C[1])/25)*x^3 - 18350080*E^((24*
C[1])/25)*x^4 - 1835008*E^((24*C[1])/25)*x^5)*#1^5 + (4194304 + 469762048*E^((24
*C[1])/25) + 1409286144*E^((24*C[1])/25)*x + 1761607680*E^((24*C[1])/25)*x^2 + 1
174405120*E^((24*C[1])/25)*x^3 + 440401920*E^((24*C[1])/25)*x^4 + 88080384*E^((2
4*C[1])/25)*x^5 + 7340032*E^((24*C[1])/25)*x^6)*#1^6 + (-33554432 - 2147483648*E
^((24*C[1])/25) - 16777216*x - 7516192768*E^((24*C[1])/25)*x - 11274289152*E^((2
4*C[1])/25)*x^2 - 9395240960*E^((24*C[1])/25)*x^3 - 4697620480*E^((24*C[1])/25)*
x^4 - 1409286144*E^((24*C[1])/25)*x^5 - 234881024*E^((24*C[1])/25)*x^6 - 1677721
6*E^((24*C[1])/25)*x^7)*#1^7 + (67108864 + 4294967296*E^((24*C[1])/25) + 6710886
4*x + 17179869184*E^((24*C[1])/25)*x + 16777216*x^2 + 30064771072*E^((24*C[1])/2
5)*x^2 + 30064771072*E^((24*C[1])/25)*x^3 + 18790481920*E^((24*C[1])/25)*x^4 + 7
516192768*E^((24*C[1])/25)*x^5 + 1879048192*E^((24*C[1])/25)*x^6 + 268435456*E^(
(24*C[1])/25)*x^7 + 16777216*E^((24*C[1])/25)*x^8)*#1^8 & , 2]^(-1))/5}, {y[x] -
> (-3*(2 + x) + Root[E^((24*C[1])/25) + (-128*E^((24*C[1])/25) - 64*E^((24*C[1])
/25)*x)*#1 + (7168*E^((24*C[1])/25) + 7168*E^((24*C[1])/25)*x + 1792*E^((24*C[1]
)/25)*x^2)*#1^2 + (-229376*E^((24*C[1])/25) - 344064*E^((24*C[1])/25)*x - 172032
*E^((24*C[1])/25)*x^2 - 28672*E^((24*C[1])/25)*x^3)*#1^3 + (4587520*E^((24*C[1])
/25) + 9175040*E^((24*C[1])/25)*x + 6881280*E^((24*C[1])/25)*x^2 + 2293760*E^((2
4*C[1])/25)*x^3 + 286720*E^((24*C[1])/25)*x^4)*#1^4 + (-58720256*E^((24*C[1])/25
) - 146800640*E^((24*C[1])/25)*x - 146800640*E^((24*C[1])/25)*x^2 - 73400320*E^(
(24*C[1])/25)*x^3 - 18350080*E^((24*C[1])/25)*x^4 - 1835008*E^((24*C[1])/25)*x^5
)*#1^5 + (4194304 + 469762048*E^((24*C[1])/25) + 1409286144*E^((24*C[1])/25)*x +
 1761607680*E^((24*C[1])/25)*x^2 + 1174405120*E^((24*C[1])/25)*x^3 + 440401920*E
^((24*C[1])/25)*x^4 + 88080384*E^((24*C[1])/25)*x^5 + 7340032*E^((24*C[1])/25)*x
^6)*#1^6 + (-33554432 - 2147483648*E^((24*C[1])/25) - 16777216*x - 7516192768*E^
((24*C[1])/25)*x - 11274289152*E^((24*C[1])/25)*x^2 - 9395240960*E^((24*C[1])/25
)*x^3 - 4697620480*E^((24*C[1])/25)*x^4 - 1409286144*E^((24*C[1])/25)*x^5 - 2348
81024*E^((24*C[1])/25)*x^6 - 16777216*E^((24*C[1])/25)*x^7)*#1^7 + (67108864 + 4
294967296*E^((24*C[1])/25) + 67108864*x + 17179869184*E^((24*C[1])/25)*x + 16777
216*x^2 + 30064771072*E^((24*C[1])/25)*x^2 + 30064771072*E^((24*C[1])/25)*x^3 + 
18790481920*E^((24*C[1])/25)*x^4 + 7516192768*E^((24*C[1])/25)*x^5 + 1879048192*
E^((24*C[1])/25)*x^6 + 268435456*E^((24*C[1])/25)*x^7 + 16777216*E^((24*C[1])/25
)*x^8)*#1^8 & , 3]^(-1))/5}, {y[x] -> (-3*(2 + x) + Root[E^((24*C[1])/25) + (-12
8*E^((24*C[1])/25) - 64*E^((24*C[1])/25)*x)*#1 + (7168*E^((24*C[1])/25) + 7168*E
^((24*C[1])/25)*x + 1792*E^((24*C[1])/25)*x^2)*#1^2 + (-229376*E^((24*C[1])/25) 
- 344064*E^((24*C[1])/25)*x - 172032*E^((24*C[1])/25)*x^2 - 28672*E^((24*C[1])/2
5)*x^3)*#1^3 + (4587520*E^((24*C[1])/25) + 9175040*E^((24*C[1])/25)*x + 6881280*
E^((24*C[1])/25)*x^2 + 2293760*E^((24*C[1])/25)*x^3 + 286720*E^((24*C[1])/25)*x^
4)*#1^4 + (-58720256*E^((24*C[1])/25) - 146800640*E^((24*C[1])/25)*x - 146800640
*E^((24*C[1])/25)*x^2 - 73400320*E^((24*C[1])/25)*x^3 - 18350080*E^((24*C[1])/25
)*x^4 - 1835008*E^((24*C[1])/25)*x^5)*#1^5 + (4194304 + 469762048*E^((24*C[1])/2
5) + 1409286144*E^((24*C[1])/25)*x + 1761607680*E^((24*C[1])/25)*x^2 + 117440512
0*E^((24*C[1])/25)*x^3 + 440401920*E^((24*C[1])/25)*x^4 + 88080384*E^((24*C[1])/
25)*x^5 + 7340032*E^((24*C[1])/25)*x^6)*#1^6 + (-33554432 - 2147483648*E^((24*C[
1])/25) - 16777216*x - 7516192768*E^((24*C[1])/25)*x - 11274289152*E^((24*C[1])/
25)*x^2 - 9395240960*E^((24*C[1])/25)*x^3 - 4697620480*E^((24*C[1])/25)*x^4 - 14
09286144*E^((24*C[1])/25)*x^5 - 234881024*E^((24*C[1])/25)*x^6 - 16777216*E^((24
*C[1])/25)*x^7)*#1^7 + (67108864 + 4294967296*E^((24*C[1])/25) + 67108864*x + 17
179869184*E^((24*C[1])/25)*x + 16777216*x^2 + 30064771072*E^((24*C[1])/25)*x^2 +
 30064771072*E^((24*C[1])/25)*x^3 + 18790481920*E^((24*C[1])/25)*x^4 + 751619276
8*E^((24*C[1])/25)*x^5 + 1879048192*E^((24*C[1])/25)*x^6 + 268435456*E^((24*C[1]
)/25)*x^7 + 16777216*E^((24*C[1])/25)*x^8)*#1^8 & , 4]^(-1))/5}, {y[x] -> (-3*(2
 + x) + Root[E^((24*C[1])/25) + (-128*E^((24*C[1])/25) - 64*E^((24*C[1])/25)*x)*
#1 + (7168*E^((24*C[1])/25) + 7168*E^((24*C[1])/25)*x + 1792*E^((24*C[1])/25)*x^
2)*#1^2 + (-229376*E^((24*C[1])/25) - 344064*E^((24*C[1])/25)*x - 172032*E^((24*
C[1])/25)*x^2 - 28672*E^((24*C[1])/25)*x^3)*#1^3 + (4587520*E^((24*C[1])/25) + 9
175040*E^((24*C[1])/25)*x + 6881280*E^((24*C[1])/25)*x^2 + 2293760*E^((24*C[1])/
25)*x^3 + 286720*E^((24*C[1])/25)*x^4)*#1^4 + (-58720256*E^((24*C[1])/25) - 1468
00640*E^((24*C[1])/25)*x - 146800640*E^((24*C[1])/25)*x^2 - 73400320*E^((24*C[1]
)/25)*x^3 - 18350080*E^((24*C[1])/25)*x^4 - 1835008*E^((24*C[1])/25)*x^5)*#1^5 +
 (4194304 + 469762048*E^((24*C[1])/25) + 1409286144*E^((24*C[1])/25)*x + 1761607
680*E^((24*C[1])/25)*x^2 + 1174405120*E^((24*C[1])/25)*x^3 + 440401920*E^((24*C[
1])/25)*x^4 + 88080384*E^((24*C[1])/25)*x^5 + 7340032*E^((24*C[1])/25)*x^6)*#1^6
 + (-33554432 - 2147483648*E^((24*C[1])/25) - 16777216*x - 7516192768*E^((24*C[1
])/25)*x - 11274289152*E^((24*C[1])/25)*x^2 - 9395240960*E^((24*C[1])/25)*x^3 - 
4697620480*E^((24*C[1])/25)*x^4 - 1409286144*E^((24*C[1])/25)*x^5 - 234881024*E^
((24*C[1])/25)*x^6 - 16777216*E^((24*C[1])/25)*x^7)*#1^7 + (67108864 + 429496729
6*E^((24*C[1])/25) + 67108864*x + 17179869184*E^((24*C[1])/25)*x + 16777216*x^2 
+ 30064771072*E^((24*C[1])/25)*x^2 + 30064771072*E^((24*C[1])/25)*x^3 + 18790481
920*E^((24*C[1])/25)*x^4 + 7516192768*E^((24*C[1])/25)*x^5 + 1879048192*E^((24*C
[1])/25)*x^6 + 268435456*E^((24*C[1])/25)*x^7 + 16777216*E^((24*C[1])/25)*x^8)*#
1^8 & , 5]^(-1))/5}, {y[x] -> (-3*(2 + x) + Root[E^((24*C[1])/25) + (-128*E^((24
*C[1])/25) - 64*E^((24*C[1])/25)*x)*#1 + (7168*E^((24*C[1])/25) + 7168*E^((24*C[
1])/25)*x + 1792*E^((24*C[1])/25)*x^2)*#1^2 + (-229376*E^((24*C[1])/25) - 344064
*E^((24*C[1])/25)*x - 172032*E^((24*C[1])/25)*x^2 - 28672*E^((24*C[1])/25)*x^3)*
#1^3 + (4587520*E^((24*C[1])/25) + 9175040*E^((24*C[1])/25)*x + 6881280*E^((24*C
[1])/25)*x^2 + 2293760*E^((24*C[1])/25)*x^3 + 286720*E^((24*C[1])/25)*x^4)*#1^4 
+ (-58720256*E^((24*C[1])/25) - 146800640*E^((24*C[1])/25)*x - 146800640*E^((24*
C[1])/25)*x^2 - 73400320*E^((24*C[1])/25)*x^3 - 18350080*E^((24*C[1])/25)*x^4 - 
1835008*E^((24*C[1])/25)*x^5)*#1^5 + (4194304 + 469762048*E^((24*C[1])/25) + 140
9286144*E^((24*C[1])/25)*x + 1761607680*E^((24*C[1])/25)*x^2 + 1174405120*E^((24
*C[1])/25)*x^3 + 440401920*E^((24*C[1])/25)*x^4 + 88080384*E^((24*C[1])/25)*x^5 
+ 7340032*E^((24*C[1])/25)*x^6)*#1^6 + (-33554432 - 2147483648*E^((24*C[1])/25) 
- 16777216*x - 7516192768*E^((24*C[1])/25)*x - 11274289152*E^((24*C[1])/25)*x^2 
- 9395240960*E^((24*C[1])/25)*x^3 - 4697620480*E^((24*C[1])/25)*x^4 - 1409286144
*E^((24*C[1])/25)*x^5 - 234881024*E^((24*C[1])/25)*x^6 - 16777216*E^((24*C[1])/2
5)*x^7)*#1^7 + (67108864 + 4294967296*E^((24*C[1])/25) + 67108864*x + 1717986918
4*E^((24*C[1])/25)*x + 16777216*x^2 + 30064771072*E^((24*C[1])/25)*x^2 + 3006477
1072*E^((24*C[1])/25)*x^3 + 18790481920*E^((24*C[1])/25)*x^4 + 7516192768*E^((24
*C[1])/25)*x^5 + 1879048192*E^((24*C[1])/25)*x^6 + 268435456*E^((24*C[1])/25)*x^
7 + 16777216*E^((24*C[1])/25)*x^8)*#1^8 & , 6]^(-1))/5}, {y[x] -> (-3*(2 + x) + 
Root[E^((24*C[1])/25) + (-128*E^((24*C[1])/25) - 64*E^((24*C[1])/25)*x)*#1 + (71
68*E^((24*C[1])/25) + 7168*E^((24*C[1])/25)*x + 1792*E^((24*C[1])/25)*x^2)*#1^2 
+ (-229376*E^((24*C[1])/25) - 344064*E^((24*C[1])/25)*x - 172032*E^((24*C[1])/25
)*x^2 - 28672*E^((24*C[1])/25)*x^3)*#1^3 + (4587520*E^((24*C[1])/25) + 9175040*E
^((24*C[1])/25)*x + 6881280*E^((24*C[1])/25)*x^2 + 2293760*E^((24*C[1])/25)*x^3 
+ 286720*E^((24*C[1])/25)*x^4)*#1^4 + (-58720256*E^((24*C[1])/25) - 146800640*E^
((24*C[1])/25)*x - 146800640*E^((24*C[1])/25)*x^2 - 73400320*E^((24*C[1])/25)*x^
3 - 18350080*E^((24*C[1])/25)*x^4 - 1835008*E^((24*C[1])/25)*x^5)*#1^5 + (419430
4 + 469762048*E^((24*C[1])/25) + 1409286144*E^((24*C[1])/25)*x + 1761607680*E^((
24*C[1])/25)*x^2 + 1174405120*E^((24*C[1])/25)*x^3 + 440401920*E^((24*C[1])/25)*
x^4 + 88080384*E^((24*C[1])/25)*x^5 + 7340032*E^((24*C[1])/25)*x^6)*#1^6 + (-335
54432 - 2147483648*E^((24*C[1])/25) - 16777216*x - 7516192768*E^((24*C[1])/25)*x
 - 11274289152*E^((24*C[1])/25)*x^2 - 9395240960*E^((24*C[1])/25)*x^3 - 46976204
80*E^((24*C[1])/25)*x^4 - 1409286144*E^((24*C[1])/25)*x^5 - 234881024*E^((24*C[1
])/25)*x^6 - 16777216*E^((24*C[1])/25)*x^7)*#1^7 + (67108864 + 4294967296*E^((24
*C[1])/25) + 67108864*x + 17179869184*E^((24*C[1])/25)*x + 16777216*x^2 + 300647
71072*E^((24*C[1])/25)*x^2 + 30064771072*E^((24*C[1])/25)*x^3 + 18790481920*E^((
24*C[1])/25)*x^4 + 7516192768*E^((24*C[1])/25)*x^5 + 1879048192*E^((24*C[1])/25)
*x^6 + 268435456*E^((24*C[1])/25)*x^7 + 16777216*E^((24*C[1])/25)*x^8)*#1^8 & , 
7]^(-1))/5}, {y[x] -> (-3*(2 + x) + Root[E^((24*C[1])/25) + (-128*E^((24*C[1])/2
5) - 64*E^((24*C[1])/25)*x)*#1 + (7168*E^((24*C[1])/25) + 7168*E^((24*C[1])/25)*
x + 1792*E^((24*C[1])/25)*x^2)*#1^2 + (-229376*E^((24*C[1])/25) - 344064*E^((24*
C[1])/25)*x - 172032*E^((24*C[1])/25)*x^2 - 28672*E^((24*C[1])/25)*x^3)*#1^3 + (
4587520*E^((24*C[1])/25) + 9175040*E^((24*C[1])/25)*x + 6881280*E^((24*C[1])/25)
*x^2 + 2293760*E^((24*C[1])/25)*x^3 + 286720*E^((24*C[1])/25)*x^4)*#1^4 + (-5872
0256*E^((24*C[1])/25) - 146800640*E^((24*C[1])/25)*x - 146800640*E^((24*C[1])/25
)*x^2 - 73400320*E^((24*C[1])/25)*x^3 - 18350080*E^((24*C[1])/25)*x^4 - 1835008*
E^((24*C[1])/25)*x^5)*#1^5 + (4194304 + 469762048*E^((24*C[1])/25) + 1409286144*
E^((24*C[1])/25)*x + 1761607680*E^((24*C[1])/25)*x^2 + 1174405120*E^((24*C[1])/2
5)*x^3 + 440401920*E^((24*C[1])/25)*x^4 + 88080384*E^((24*C[1])/25)*x^5 + 734003
2*E^((24*C[1])/25)*x^6)*#1^6 + (-33554432 - 2147483648*E^((24*C[1])/25) - 167772
16*x - 7516192768*E^((24*C[1])/25)*x - 11274289152*E^((24*C[1])/25)*x^2 - 939524
0960*E^((24*C[1])/25)*x^3 - 4697620480*E^((24*C[1])/25)*x^4 - 1409286144*E^((24*
C[1])/25)*x^5 - 234881024*E^((24*C[1])/25)*x^6 - 16777216*E^((24*C[1])/25)*x^7)*
#1^7 + (67108864 + 4294967296*E^((24*C[1])/25) + 67108864*x + 17179869184*E^((24
*C[1])/25)*x + 16777216*x^2 + 30064771072*E^((24*C[1])/25)*x^2 + 30064771072*E^(
(24*C[1])/25)*x^3 + 18790481920*E^((24*C[1])/25)*x^4 + 7516192768*E^((24*C[1])/2
5)*x^5 + 1879048192*E^((24*C[1])/25)*x^6 + 268435456*E^((24*C[1])/25)*x^7 + 1677
7216*E^((24*C[1])/25)*x^8)*#1^8 & , 8]^(-1))/5}}

Maple raw input

dsolve((6+3*x+5*y(x))*diff(y(x),x) = 2+x+7*y(x), y(x),'implicit')

Maple raw output

-4/3*ln((-y(x)+2+x)/(2+x))+1/3*ln((-5*y(x)-2-x)/(2+x))-ln(2+x)-_C1 = 0